Hello linalg. Welcome to PF !linalg said:Use the dimension theorem to show that every polynomial p(x) in Pn can be written in the form p(x)=q(x+1)-q(x) for some polynomial q(x) in Pn+1.
I need to see all the steps so that I understand how to do it.
PLease and Thank you
This isn't how it words here at Physics Forums.linalg said:I need to see all the steps so that I understand how to do it.
What question? You have not asked a single question.linalg said:Ok Mark44 just answer the question then
The Dimension Theorem, also known as the Dimension Formula, is a mathematical theorem that states the dimension of any vector space is equal to the number of vectors in a basis for that space.
The Dimension Theorem can be used to prove that polynomials can be written using a basis. This means that any polynomial can be expressed as a linear combination of a finite number of basis polynomials.
A basis for polynomials is a set of polynomials that can be used to create any other polynomial through linear combinations. A common basis for polynomials is the set of all monomials of a certain degree.
The Dimension Theorem states that the dimension of a vector space is equal to the number of vectors in the basis. Since the space of polynomials of degree n has dimension n+1, this means that any polynomial of degree n can be expressed as a linear combination of n+1 basis polynomials. Therefore, polynomials can be written using a basis.
Yes, the Dimension Theorem can be applied to any vector space, not just polynomials. It can also be used to prove the existence of a basis for any vector space.